In the great theater of the American economy, it all seems so straightforward. Enter, stage right, a group called ''the private sector.'' Enter, stage left, another known as ''government.'' The orchestra strikes up, and off goes the cast - improvising its way toward something called the gross national product.
But look hard, and a third bunch appears: ''the nonprofits.'' They represent a galaxy of charitable, social, educational, and cultural organizations, loosely attached to the private foundations that fund them. Ill at ease as the play unfolds, they don't quite know where to stand - above, between, or behind the prima donnas. A modest group, it goes by many names: the Third Sector, the Independent Sector, the Invisible Sector, the Sector Nobody Knows.
The latter term - coined by Waldemar Nielsencq in a book entitled ''The Endangered Sector'' - is not strictly accurate. The nonprofit sector is part of daily life, standing behind everything from YMCAs to art museums, community day-care centers to public universities. Yet few people recognize the full import of this third sector - nor the extent of the challenges it's facing.
How important is it? One measure is its size. Estimates made by Independent Sector (a Washington-based umbrella organization for nonprofits) pegged the total income of these organizations in 1981 at $192 billion - more than the currently projected federal deficit. The figure includes some $64.5 billion worth of time contributed by corporate executives and other volunteers.
America's philanthropic foundations - Ford, Carnegie, Mellon, and a host of others - are no small part of this picture. Figures published last week by the Foundation Center show a total of 21,967 such foundations, with combined assets of $47.6 billion and an annual outlay of gifts totaling $3.8 billion. Owning perhaps 10 percent of all property in the country (according to Nielsencq), they provide employment for a similar percentage of the work force.
So where, in all that affluence, are the challenges?
* Inflation. Between 1972 and 1979, spending by major foundations rose 60 percent. But as the dollar spluttered, the value of that outlay actually declined by 8 percent.Since then, inflation has eased - although grantmaking agencies have yet to catch up with the losses of those years.
* Federal laws. In the decade before the 1969 Tax Reform Act, 837 major foundations were created. But from 1970 to 1979, only 334 new ones were added; and from 1980 to '82, only 29. Not surprisingly, the total number of foundations tallied in the Foundation Center's recent report (21,967) has fallen by more than 500 from its previous level of 22,535.
* Shift in federal priorities. But the largest challenge - and the one least understood by the public, according to former Stanford University president Richard W. Lyman - comes from federal budget cutbacks in recent years. Dr. Lyman, president of the Rockefeller Foundation and chairman of Independent Sector, likes to call attention to a little-known fact: that US government funding for nonprofit organizations considerably outweighs contributions by private donors. ''Government,'' he told a conference held by the Associated Grantmakers of Massachusetts recently, has ''come to dominate as never before - to be the fat boy in the canoe.''
As a result, the depth and suddenness of recent budget cuts have radically shifted the equilibrium of the nonprofits. With federal funding slashed for everything from free soup to free verse, third-sector proponents are searching for new ways forward. They can't simply fill in where federal agencies have backed out: The foundations aren't that big. But neither can they go on as before.
What is needed, says Dr. Lyman, is ''serious discourse'' about the ways public and private funders can cooperate. And that, he laments, is exactly what is lacking. ''We still hear far too much argument that is simply - even simple-mindedly - for or against government involvement,'' he says, adding that ''freedom and compassion are still fought over as if they were somehow at odds with one another.'' What he calls ''the tricky and delicate business'' of fostering ''combined operations between government and private enterprise'' is, he says, ''too seldom on the political agenda.''
In a nation that cut its teeth on public-private relationships - from revolutionary Minutemen to the volunteer fire departments - these are words to take to heart. Trotting out pat conceptions of ''good'' government and ''bad'' business - and vice versa - simply won't work. On America's economic stage, neither of those two parties can carry the full weight of the production by itself. It is time, now, to call forward that third presence, cut through the simple-mindedness, and write for it a part worthy of its considerable talents.